Rút gọn P

To simplify the expression, we can rationalize the denominators of the fractions.

First, let's rationalize the denominator of the first fraction:
Multiply the numerator and denominator by the conjugate of the denominator, which is √(a+2):
((a+4) * √(a+2)) / ((√(a+2)) * (√(a+2))) = (a+4)√(a+2) / (a+2)

Next, let's rationalize the denominator of the second fraction:
Multiply the numerator and denominator by the conjugate of the denominator, which is 2+√a:
((4-a) * (2+√a)) / ((2+√a) * (2+√a)) = (4-a)(2+√a) / (2+√a)^2 = (4-a)(2+√a) / (4+2√a+2√a+a) = (4-a)(2+√a) / (a+4+4√a)

Now, the expression becomes:
[(a+4)√(a+2) / (a+2)] + [(4-a)(2+√a) / (a+4+4√a)]

To simplify further, we can combine the fractions by finding a common denominator:
[(a+4)√(a+2)(a+4+4√a) + (4-a)(2+√a)(a+2)] / [(a+2)(a+4+4√a)]

Expanding and simplifying the numerator:
[(a+4)(a+4+4√a)√(a+2) + (4-a)(2+√a)(a+2)] / [(a+2)(a+4+4√a)]

[(a^2+8a+16+16√a(a+1))√(a+2) + (8+2√a-2a-√a^2)(a+2)] / [(a+2)(a+4+4√a)]

[(a^2+8a+16+16√a(a+1))√(a+2) + (8+2√a-2a-a)(a+2)] / [(a+2)(a+4+4√a)]

[(a^2+8a+16+16√a(a+1))√(a+2) + (8+√a-3a)(a+2)] / [(a+2)(a+4+4√a)]

[(a^2+8a+16+16√a(a+1))√(a+2) + (8+√a-3a)(a+2)] / [(a+2)(a+4+4√a)]

Simplifying the denominator:
(a+2)(a+4+4√a) = a^2+6a+8+4a√a+4√a+16 = a^2+10a+24+4a√a

Finally, the expression simplifies to:
[(a^2+8a+16+16√a(a+1))√(a+2) + (8+√a-3a)(a+2)] / (a^2+10a+24+4a√a)